Quiz — 6589493253937_-BAC EXAM CORRECTION_ Done.pdf

Question 1 sur 10 20:00

[{"id":26574,"question":"Quelle est la dérivée de la fonction f(x) = ln(3x² + 1) ?","option_a":"A) (6x)\/(3x² + 1)","option_b":"B) (3x)\/(3x² + 1)","option_c":"C) (6x²)\/(3x² + 1)","option_d":"D) (6)\/(3x² + 1)","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de ln(u) est u'\/u. Ici, u = 3x² + 1, donc u' = 6x. Ainsi, f'(x) = 6x\/(3x² + 1).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"A) (6x)\/(3x² + 1)\", \"b\": \"B) (3x)\/(3x² + 1)\", \"c\": \"C) (6x²)\/(","_debug_options_count":4},{"id":26575,"question":"Une suite géométrique de raison q = -2 et de premier terme u₀ = 3 est-elle convergente ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Une suite géométrique converge si et seulement si |q| \u003C 1. Ici, |q| = 2 \u003E 1, donc la suite diverge.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":26576,"question":"Quelle est l'équation réduite de la tangente à la courbe y = x² - 4x + 5 au point d'abscisse x = 3 ?","option_a":"A) y = 2x - 3","option_b":"B) y = 2x - 7","option_c":"C) y = -2x + 15","option_d":"D) y = 2x + 1","option_e":"","option_f":"","bonne_reponse":"b","explication":"La tangente en x = 3 a pour équation y = f'(3)(x - 3) + f(3). f'(x) = 2x - 4, donc f'(3) = 2. f(3) = 9 - 12 + 5 = 2. Ainsi, y = 2(x - 3) + 2 = 2x - 4.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"A) y = 2x - 3\", \"b\": \"B) y = 2x - 7\", \"c\": \"C) y = -2x + 15\", \"d\"","_debug_options_count":4},{"id":26577,"question":"La probabilité conditionnelle P(A|B) est-elle égale à P(A ∩ B)\/P(B) ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Par définition, P(A|B) = P(A ∩ B)\/P(B) si P(B) ≠ 0. Cette formule est fondamentale en probabilités.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":26578,"question":"Quelle est la solution générale de l'équation différentielle y' + 2y = 0 ?","option_a":"A) y = Ce^(2x)","option_b":"B) y = Ce^(-2x)","option_c":"C) y = Ce^(x\/2)","option_d":"D) y = Ce^(-x\/2)","option_e":"","option_f":"","bonne_reponse":"b","explication":"L'équation y' + 2y = 0 est une équation différentielle linéaire du premier ordre. Sa solution générale est y = Ce^(-2x).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"A) y = Ce^(2x)\", \"b\": \"B) y = Ce^(-2x)\", \"c\": \"C) y = Ce^(x\/2)\", ","_debug_options_count":4},{"id":26579,"question":"Le produit scalaire de deux vecteurs orthogonaux est-il toujours nul ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Par définition, deux vecteurs sont orthogonaux si et seulement si leur produit scalaire est nul.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":26580,"question":"Quelle est l'intégrale de la fonction f(x) = e^(-x) entre 0 et +∞ ?","option_a":"A) 0","option_b":"B) 1","option_c":"C) -1","option_d":"D) +∞","option_e":"","option_f":"","bonne_reponse":"b","explication":"L'intégrale impropre ∫₀^∞ e^(-x) dx = [-e^(-x)]₀^∞ = 0 - (-1) = 1.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"A) 0\", \"b\": \"B) 1\", \"c\": \"C) -1\", \"d\": \"D) +∞\"}}","_debug_options_count":4},{"id":26581,"question":"Une matrice carrée A est inversible si et seulement si son déterminant est strictement positif.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Une matrice est inversible si et seulement si son déterminant est non nul (≠ 0), pas forcément positif.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":26582,"question":"Quelle est la limite de la suite uₙ = (n² + 3n)\/(2n² - 5) quand n tend vers +∞ ?","option_a":"A) 0","option_b":"B) 1\/2","option_c":"C) 1","option_d":"D) +∞","option_e":"","option_f":"","bonne_reponse":"c","explication":"En divisant numérateur et dénominateur par n², on obtient uₙ = (1 + 3\/n)\/(2 - 5\/n²). La limite quand n→∞ est 1\/2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"A) 0\", \"b\": \"B) 1\/2\", \"c\": \"C) 1\", \"d\": \"D) +∞\"}}","_debug_options_count":4},{"id":26583,"question":"La fonction f(x) = x³ - 3x² + 2x est-elle croissante sur l'intervalle [0 ; 2] ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La dérivée f'(x) = 3x² - 6x + 2. Sur [0 ; 2], f'(x) = 0 a deux solutions, et f'(x) \u003C 0 sur une partie de l'intervalle. La fonction n'est pas croissante sur tout [0 ; 2].","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4}]